No, they wouldn't do something like that and not make you pay for it. It's probably to make sure that you grew up in an English speaking country. There are a lot of people with passports from the countries they accept who didn't grow up there.

I think most passports indicate if you were born a citizen or naturalized, so that may not be exactly why they do it, but the first thing you should know about Korea is that there's not always a good (or any) reason for the way things are done. Maybe ten years ago some Korean guy who couldn't speak English copied and pasted an application to lease an apartment when he was put in charge of drafting the EPIK application and it has remained through the years.

I've known people who have had criminal records in Korea who have gotten hired because they don't even bother to check those. You can live in Korea for years and get in all kinds of trouble, and still, all you need is a clean FBI background check.